(2) Another class of examples of total integrals arises from the
graded case. Recall from [1]} that let \(G\)  be a finite group and \(H=kG\)  be the monoidal Hom-Hopf algebra with structures:
\(&&xy=\alpha (xy), 1x=\alpha (x),\varepsilon (x)=1\\&&\Delta (x)=\alpha ^{-1}(x)\otimes \alpha ^{-1}(x),~x,y\in G.\) 
